Twitter seems to be making it harder and harder to pull your own content out of their ecosystem.
They recently changed their API over to version 1.1, and now require all requests to be authenticated using OAuth. Even simple GET requests need OAuth and setting up it is a pain, if all you want to do is display a list of tweets.
The simplest way to display a list of tweets is to use Twitter’s embedded timelines. Not only can you use them to display tweets, but you can also use them to display search results, favorites and lists.
drush dl twitter_block